It’s feasible to use HTTP-01 validation, which uses a request to e.g. http://www.example.com/.well-known/acme-challenge/[random filename] if you configure your other web servers to respond appropriately, by redirecting or reverse proxying to your certificate server, or by having the certificate server copy files to them, or using NFS or something.

There’s also the question of how to deploy new certificates (and keys) to your other servers. You can write a Certbot hook to do it. The client GetSSL has a built-in feature to copy them over SSH (and to perform validation that way).